Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms

Abstract

In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*} \begin{cases} \displaystyle - \Delta_1 u= h(u)f & \text{in } \Omega, \\ \newline u=0 & \text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(\Omega)$ with $m\geq 1$ is a nonnegative function and $h\colon \mathbb{R}^+ \to \mathbb{R}^+$ is continuous, possibly singular at the origin and bounded at infinity. Without any growth restrictions on $h$ at zero, we prove existence of global finite energy solutions in $BV(\Omega)$ under sharp conditions on the summability of $f$ and on the behaviour of $h$ at infinity. Roughly speaking, the faster $h$ goes to zero at infinity, the less regularity is required on $f$. In contrast to the $p$-Laplacian case ($p>1$), we show that the behaviour of $h$ at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna (\cite{lm}) to the case of the $1$-Laplacian as principal operator.